pwm2lmom {lmomco} | R Documentation

## Probability-Weighted Moments to L-moments

### Description

Converts the Probability-Weighted Moments (PWM) to the L-moments given the PWM. The conversion is linear so procedures based on PWMs and identical to those based on L-moments.

λ_1 = β_0 mbox{,}

λ_2 = 2β_1 - β_0 mbox{,}

λ_3 = 6β_2 - 6β_1 + β_0 mbox{,}

λ_4 = 20β_3 - 30β_2 + 12β_1 - β_0 mbox{,}

λ_5 = 70β_4 - 140β_3 + 90β_2 - 20β_1 + β_0 mbox{,}

tau = λ_2/λ_1 mbox{,}

tau_3 = λ_3/λ_2 mbox{,}

tau_4 = λ_4/λ_2 mbox{, and}

tau_5 = λ_5/λ_2 mbox{.}

The general expression and the expression used for computation if the argument is a vector of PWMs is

λ_{r+1} = sum^r_{k=0} (-1)^{r-k}{r choose k}{r+k choose k} β_{k+1}

### Usage

    pwm2lmom(pwm)

### Arguments

`pwm` | A PWM object created by `pwm.ub` or similar.

### Details

The Probability Weighted Moments (PWMs) are linear combinations of the L-moments and therefore contain the same statistical information of the data as the L-moments. However, the PWMs are harder to interpret as measures of probability distributions. The linearity between L-moments and PWMs means that procedures base on one are equivalent to the other.

The function can take a variety of PWM argument types in `pwm`. The function checks whether the argument is a list and if so attempts to extract the β_r's from `list` names such as `BETA0`, `BETA1`, and so on. If the extraction is successful, then a list of L-moments similar to `lmom.ub` is returned. If the extraction was not successful, then a `list` name `betas` is checked; if `betas` is found then this vector of PWMs is used to compute the L-moments. If `pwm` is a `list` but can not be routed in the function, a `warnings` is made and `NULL` returned. If the `pwm` argument is a `vector`, then this vector of PWMs is used to compute the L-moments are returned.

### Value

One of two R `list` are returned.

`L1` | Arithmetic mean
`L2` | L-scale—analogous to standard deviation
`LCV` | coefficient of L-variation—analogous to coe. of variation
`TAU3` | The third L-moment ratio or L-skew—analogous to skew
`TAU4` | The fourth L-moment ratio or L-kurtosis—analogous to kurtosis
`TAU5` | The fifth L-moment ratio
`L3` | The third L-moment
`L4` | The fourth L-moment
`L5` | The fifth L-moment newline newline or the following list newline mbox{ }
`lambdas` | The L-moments
`ratios` | The L-moment ratios
`source` | Source of the L-moments (`pwm2lmom`)

### Author(s)

W.H. Asquith

### References

Greenwood, J.A., Landwehr, J.M., Matalas, N.C., and Wallis, J.R., 1979, Probability weighted moments—Definition and relation to parameters of several distributions expressable in inverse form: Water Resources Research, vol. 15, p. 1,049–1,054.

Hosking, J.R.M., 1990, L-moments–Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, vol. 52, p. 105–124.

Hosking, J.R.M. and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

### See Also

`lmom.ub`, `pwm.ub`, `pwm`, `lmom2pwm`

### Examples

    D <- c(123,34,4,654,37,78)
    pwm2lmom(pwm.ub(D))
    pwm2lmom(pwm(D))

    pwm2lmom(pwm(rnorm(100)))

[Package lmomco version 0.92 Index]